FRG: Collaborative Research: Arithmetic and equidistribution on homogeneous spaces

FRG：协作研究：齐次空间上的算术和等分布

• 批准号: 0903110
• 负责人: Akshay Venkatesh
• 金额: $ 8.35万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0903110&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Arithmetic; equidistribution

In recent years, it has become clear that many interesting problems, in particular problems in arithmetic, quantum chaos and the theory of L-functions, may be profitably reduced to questions concerning equidistribution of points or measures on homogeneous spaces. These questions regarding equidistribution can be approached from many angles. Two theories which have proved to be particularly well-suited to study such questions are the spectral theory of automorphic forms, which is closely related to the theory of L-functions, and the theory of dynamical systems, particularly the study of unipotent and more general flows on these homogeneous spaces. Recently there has been considerable progress involving tools such as special value formulae for L-functions, and (partial) classification results for measures invariant under higher rank torus actions. Particularly exciting is the possibility, already realized in some instances, of combining these techniques. The purpose of the proposed FRG is to investigate further this circle of ideas, which we believe has the potential to impact many other problems related to the above. The result of these investigations will be a deeper understanding of the dynamics of group actions on homogeneous spaces, of the analytic theory of automorphic forms, and the (sometimes unexpected) applications to problems ofarithmetic nature.The present project is concerned with a surprising link between two classical fields of mathematics of quite disparate origin: number theory and dynamics. The study of number theory began thousands of years ago, motivated, in significant part, by questions about prime numbers. On the other hand, ergodic theory and dynamics are mathematical fields of more recent provenance, which arose from studying the long-term evolution of complicated deterministic processes -- such as planetary motion. It is a striking fact (which has only recently begun to be heavily exploited) that, in certain contexts, ideas from ergodic theory interact very deeply with classical problems in number theory. This project will enhance our understanding of this inter-relation and how we can combine knowledge from both of these fruitful disciplines effectively.

近年来，很明显，许多有趣的问题，尤其是算术，量子混乱和L功能理论的问题，可能会降低到有关均匀空间的分数或措施的问题。这些有关等距的问题可以从许多角度解决。事实证明，这两种特别适合研究此类问题的理论是自动形式的光谱理论，它们与L功能理论密切相关，而动力学系统的理论，尤其是对这些同质空间的单位和更一般流动的研究。最近，涉及工具的进展取得了很大的进步，例如l功能的特殊价值公式，以及（部分）在较高级别的圆环动作下不变的措施的分类结果。特别令人兴奋的是，在某些情况下已经实现了结合这些技术的可能性。拟议的FRG的目的是进一步研究这一思想循环，我们认为这有可能影响与上述有关的许多其他问题。这些调查的结果将是对群体行动对同质空间的动态的更深入的了解，对自动形式的分析理论以及（有时意外的）对Arithmetication性质问题的应用（有时意外）的应用。目前的项目与非常平差的来源数学的数学数学领域之间的两个经典领域之间的令人惊讶的联系：数字理论：数字理论和动力学。数字理论的研究始于数千年前，在很大程度上是由于有关质数的问题而动机。 另一方面，千古理论和动力学是最新出处的数学领域，这是由于研究复杂的确定性过程的长期演变而引起的，例如行星运动。这是一个惊人的事实（直到最近才开始被大量利用），在某些情况下，来自千古理论的思想与数字理论中的经典问题非常深入。 该项目将增强我们对这一相互关系的理解，以及我们如何有效地结合这两个富有成果的学科的知识。